where \(\sigma _{pp}\)  is the inelastic \(pp\)  cross section, with an approximate expression for its energy dependence being \(\sigma _{pp}\simeq (34.9+1.985L+0.18L^2)\)  mb, with \(L={\rm ln}(E_p/{\rm TeV})\)  [1]}. The quantity \(D\)  is the distance traversed by the protons through the target gas, so that \(N_{\rm H}\equiv n_{\rm H}D\)  is the associated column density, and we have assumed that the source medium is thin, i.e. that \(\sigma _{pp}N_{\rm H}\ll 1\) . Given that \(\sigma _{pp}\sim 10^{-25}\)  cm\(^2\) , the source will be thin as long as \(N_{\rm H}\ll 10^{25}\)  cm\(^{-2}\) . The function \(F_{\nu _i}\) , depending on \(x_\nu \equiv E_\nu /E_p\)  and on \(E_p\) , describes the neutrino yield in the \(pp\)  interaction, and is in principle different for the different neutrino and antineutrino flavors. At low energies there is a noticeable excess of positive pions being produced in \(pp\)  interactions, leading in particular to a net excess of \(\nu _e\)  over \(\bar{\nu }_e\) . However, when considering very high energies it is a good approximation to assume that the \(\pi ^{+}\)  and \(\pi ^{-}\)  multiplicities are similar. Hence, for the energies that we will explore here the fluxes of neutrinos and antineutrinos are expected to be similar for the \(pp\)  scenarios, so that we will consider for each flavor the sum of neutrinos and antineutrinos that were obtained in [2]}, and just refer to them as neutrinos. Note also that neutrino telescopes, such as IceCube, cannot distinguish neutrinos from antineutrinos, with the rare exception of the \(\bar{\nu }_e\)  induced \(W\) -resonance events at 6.3 PeV neutrino energies resulting from the interaction with atomic electrons (\(\bar{\nu }_e e\rightarrow W\rightarrow anything\) ). The index \(i\)  in Eq. (REF ) identifies the type of neutrino produced, with \(\nu _\mu ^{(1)}\)  being those from the pion decay, while \(\nu _\mu ^{(2)}\)  and \(\nu _e\)  those from the subsequent muon decay.
The spectrum of \(\nu _e\)  turns out to be, to an accuracy better than 5%, equal to that of \(\nu _\mu ^{(2)}\) , and hence the two fluxes are just assumed to be the same. To compute these functions one has to account for the fact that the muon produced in the pion decay is polarized [3]}, since this affects the energy distribution of its decay products.
These functions have been parameterized
in [2]} using simulations based on the SIBYLL 2.1 code to account for the details of the \(pp\)  hadronic interactions (finding comparable results also for simulations based on the QGSJet hadronic interaction model). There is also a small contribution to the neutrino fluxes from charged kaon decays (which is below the 10% level, see e.g. [5]}), but following [2]} we do not include it since this should not affect significantly our results.
